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Register a new userA Large deviation principle for last passage times in an asymmetric Bernoulli potential
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We prove a large deviation principle and give an expression for the rate function, for the last passage time in a Bernoulli environment. The model is exactly solvable and its invariant version satisfies a Burke-type property. Finally, we compute explicit limiting logarithmic moment generating functions for both the classical and the invariant models. The shape function of this model exhibits a flat edge in certain directions, and we also discuss the rate function and limiting log-moment generating functions in those directions.
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We study the sequence alignment problem and its independent version, the discrete Hammersley process with an exploration penalty. We obtain rigorous upper bounds for the number of optimality regions in both models near the soft edge. At zero penalty the independent model becomes an exactly solvable model and we identify cases for which the law of the last passage time converges to a Tracy-Widom law.
We initiate a study of large deviations for block model random graphs in the dense regime. Following Chatterjee-Varadhan(2011), we establish an LDP for dense block models, viewed as random graphons. As an application of our result, we study upper tail large deviations for homomorphism densities of regular graphs. We identify the existence of a symmetric phase, where the graph, conditioned on the rare event, looks like a block model with the same block sizes as the generating graphon. In specific examples, we also identify the existence of a symmetry breaking regime, where the conditional structure is not a block model with compatible dimensions. This identifies a reentrant phase transition phenomenon for this problem---analogous to one established for Erdos-Renyi random graphs (Chatterjee-Dey(2010), Chatterjee-Varadhan(2011)). Finally, extending the analysis of Lubetzky-Zhao(2015), we identify the precise boundary between the symmetry and symmetry breaking regime for homomorphism densities of regular graphs and the operator norm on Erdos-Renyi bipartite graphs.
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